In this paper, we consider a multiple-input multiple-output (MIMO) radar system for localizing a target based on its reflected echo signals. Specifically, we aim to estimate the random and unknown angle information of the target, by exploiting its prior distribution information. First, we characterize the estimation performance by deriving the posterior Cram\'er-Rao bound (PCRB), which quantifies a lower bound of the estimation mean-squared error (MSE). Since the PCRB is in a complicated form, we derive a tight upper bound of it to approximate the estimation performance. Based on this, we analytically show that by exploiting the prior distribution information, the PCRB is always no larger than the Cram\'er-Rao bound (CRB) averaged over random angle realizations without prior information exploitation. Next, we formulate the transmit signal optimization problem to minimize the PCRB upper bound. We show that the optimal sample covariance matrix has a rank-one structure, and derive the optimal signal solution in closed form. Numerical results show that our proposed design achieves significantly improved PCRB performance compared to various benchmark schemes.
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Evolutionary differential equation discovery proved to be a tool to obtain equations with less a priori assumptions than conventional approaches, such as sparse symbolic regression over the complete possible terms library. The equation discovery field contains two independent directions. The first one is purely mathematical and concerns differentiation, the object of optimization and its relation to the functional spaces and others. The second one is dedicated purely to the optimizational problem statement. Both topics are worth investigating to improve the algorithm's ability to handle experimental data a more artificial intelligence way, without significant pre-processing and a priori knowledge of their nature. In the paper, we consider the prevalence of either single-objective optimization, which considers only the discrepancy between selected terms in the equation, or multi-objective optimization, which additionally takes into account the complexity of the obtained equation. The proposed comparison approach is shown on classical model examples -- Burgers equation, wave equation, and Korteweg - de Vries equation.
The $L_{\infty}$ star discrepancy is a measure for the regularity of a finite set of points taken from $[0,1)^d$. Low discrepancy point sets are highly relevant for Quasi-Monte Carlo methods in numerical integration and several other applications. Unfortunately, computing the $L_{\infty}$ star discrepancy of a given point set is known to be a hard problem, with the best exact algorithms falling short for even moderate dimensions around 8. However, despite the difficulty of finding the global maximum that defines the $L_{\infty}$ star discrepancy of the set, local evaluations at selected points are inexpensive. This makes the problem tractable by black-box optimization approaches. In this work we compare 8 popular numerical black-box optimization algorithms on the $L_{\infty}$ star discrepancy computation problem, using a wide set of instances in dimensions 2 to 15. We show that all used optimizers perform very badly on a large majority of the instances and that in many cases random search outperforms even the more sophisticated solvers. We suspect that state-of-the-art numerical black-box optimization techniques fail to capture the global structure of the problem, an important shortcoming that may guide their future development. We also provide a parallel implementation of the best-known algorithm to compute the discrepancy.
We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$'th and the $k+1$'th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson's stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.
Due to the nature of pure-tone audiometry test, hearing loss data often has a complicated correlation structure. Generalized estimating equation (GEE) is commonly used to investigate the association between exposures and hearing loss, because it is robust to misspecification of the correlation matrix. However, this robustness typically entails a moderate loss of estimation efficiency in finite samples. This paper proposes to model the correlation coefficients and use second-order generalized estimating equations to estimate the correlation parameters. In simulation studies, we assessed the finite sample performance of our proposed method and compared it with other methods, such as GEE with independent, exchangeable and unstructured correlation structures. Our method achieves an efficiency gain which is larger for the coefficients of the covariates corresponding to the within-cluster variation (e.g., ear-level covariates) than the coefficients of cluster-level covariates. The efficiency gain is also more pronounced when the within-cluster correlations are moderate to strong, or when comparing to GEE with an unstructured correlation structure. As a real-world example, we applied the proposed method to data from the Audiology Assessment Arm of the Conservation of Hearing Study, and studied the association between a dietary adherence score and hearing loss.
This work was originally published by the author in 1999 in a book [1] and later became part of the author's doctoral thesis in 1999 [2]. Since the original language of these works is not English, the author provides a translation of the key ideas of these publications in this work. In addition, the chapter related to numerical experiments was recalculated on modern computers and using contemporary benchmark datasets. This article presents a novel approach to solving Hartree-Fock equations using Toeplitz and tensor matrices and bases based on regular finite elements. The issues discussed include the choice of basis, the dependence of data volume and number of arithmetic operations on the number of basis functions, as well as the arithmetic complexity and accuracy of computing two- and four-center integrals. The approach has been implemented in a software package, and results have been obtained that are in good agreement with theory.
The issue of ensuring privacy for users who share their personal information has been a growing priority in a business and scientific environment where the use of different types of data and the laws that protect it have increased in tandem. Different technologies have been widely developed for static publications, i.e., where the information is published only once, such as k-anonymity and {\epsilon}-differential privacy. In the case where microdata information is published dynamically, although established notions such as m-invariance and {\tau}-safety already exist, developments for improving utility remain superficial. We propose a new heuristic approach for the NP-hard combinatorial problem of m-invariance and {\tau}-safety, which is based on a mathematical optimization column generation scheme. The quality of a solution to m-invariance and {\tau}-safety can be measured by the Information Loss (IL), a value in [0,100], the closer to 0 the better. We show that our approach improves by far current heuristics, providing in some instances solutions with ILs of 1.87, 8.5 and 1.93, while the state-of-the art methods reported ILs of 39.03, 51.84 and 57.97, respectively.
This paper investigates the energy efficiency of a multiple-input multiple-output (MIMO) integrated sensing and communications (ISAC) system, in which one multi-antenna base station (BS) transmits unified ISAC signals to a multi-antenna communication user (CU) and at the same time use the echo signals to estimate an extended target. We focus on one particular ISAC transmission block and take into account the practical on-off non-transmission power at the BS. Under this setup, we minimize the energy consumption at the BS while ensuring a minimum average data rate requirement for communication and a maximum Cram\'er-Rao bound (CRB) requirement for target estimation, by jointly optimizing the transmit covariance matrix and the ``on'' duration for active transmission. We obtain the optimal solution to the rate-and-CRB-constrained energy minimization problem in a semi-closed form. Interestingly, the obtained optimal solution is shown to unify the spectrum-efficient and energy-efficient communications and sensing designs. In particular, for the special MIMO sensing case with rate constraint inactive, the optimal solution follows the isotropic transmission with shortest ``on'' duration, in which the BS radiates the required sensing energy by using sufficiently high power over the shortest duration. For the general ISAC case, the optimal transmit covariance solution is of full rank and follows the eigenmode transmission based on the communication channel, while the optimal ``on'' duration is determined based on both the rate and CRB constraints. Numerical results show that the proposed ISAC design achieves significantly reduced energy consumption as compared to the benchmark schemes based on isotropic transmission, always-on transmission, and sensing or communications only designs, especially when the rate and CRB constraints become stringent.
We propose a novel generative saliency prediction framework that adopts an informative energy-based model as a prior distribution. The energy-based prior model is defined on the latent space of a saliency generator network that generates the saliency map based on a continuous latent variables and an observed image. Both the parameters of saliency generator and the energy-based prior are jointly trained via Markov chain Monte Carlo-based maximum likelihood estimation, in which the sampling from the intractable posterior and prior distributions of the latent variables are performed by Langevin dynamics. With the generative saliency model, we can obtain a pixel-wise uncertainty map from an image, indicating model confidence in the saliency prediction. Different from existing generative models, which define the prior distribution of the latent variables as a simple isotropic Gaussian distribution, our model uses an energy-based informative prior which can be more expressive in capturing the latent space of the data. With the informative energy-based prior, we extend the Gaussian distribution assumption of generative models to achieve a more representative distribution of the latent space, leading to more reliable uncertainty estimation. We apply the proposed frameworks to both RGB and RGB-D salient object detection tasks with both transformer and convolutional neural network backbones. We further propose an adversarial learning algorithm and a variational inference algorithm as alternatives to train the proposed generative framework. Experimental results show that our generative saliency model with an energy-based prior can achieve not only accurate saliency predictions but also reliable uncertainty maps that are consistent with human perception. Results and code are available at \url{//github.com/JingZhang617/EBMGSOD}.
Separating signals from an additive mixture may be an unnecessarily hard problem when one is only interested in specific properties of a given signal. In this work, we tackle simpler "statistical component separation" problems that focus on recovering a predefined set of statistical descriptors of a target signal from a noisy mixture. Assuming access to samples of the noise process, we investigate a method devised to match the statistics of the solution candidate corrupted by noise samples with those of the observed mixture. We first analyze the behavior of this method using simple examples with analytically tractable calculations. Then, we apply it in an image denoising context employing 1) wavelet-based descriptors, 2) ConvNet-based descriptors on astrophysics and ImageNet data. In the case of 1), we show that our method better recovers the descriptors of the target data than a standard denoising method in most situations. Additionally, despite not constructed for this purpose, it performs surprisingly well in terms of peak signal-to-noise ratio on full signal reconstruction. In comparison, representation 2) appears less suitable for image denoising. Finally, we extend this method by introducing a diffusive stepwise algorithm which gives a new perspective to the initial method and leads to promising results for image denoising under specific circumstances.
Recent advances in maximizing mutual information (MI) between the source and target have demonstrated its effectiveness in text generation. However, previous works paid little attention to modeling the backward network of MI (i.e., dependency from the target to the source), which is crucial to the tightness of the variational information maximization lower bound. In this paper, we propose Adversarial Mutual Information (AMI): a text generation framework which is formed as a novel saddle point (min-max) optimization aiming to identify joint interactions between the source and target. Within this framework, the forward and backward networks are able to iteratively promote or demote each other's generated instances by comparing the real and synthetic data distributions. We also develop a latent noise sampling strategy that leverages random variations at the high-level semantic space to enhance the long term dependency in the generation process. Extensive experiments based on different text generation tasks demonstrate that the proposed AMI framework can significantly outperform several strong baselines, and we also show that AMI has potential to lead to a tighter lower bound of maximum mutual information for the variational information maximization problem.
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